Since the literature for Keller-Segel models with a reaction term is vast, we will present some of the most important results. System (REF ) with \(f(u) \equiv 0,\ m=\gamma =1\)  is the classical parabolic-elliptic Keller-Segel model which expresses the random movement(brownian motion) of the cells with a bias directed by the chemoattractant concentration [1]} see also [2]}, [3]}, [4]} and the references therein. Moreover, for
\(& u_t=\Delta u-\chi \nabla \cdot (u \nabla c), \\& -\Delta c=u-1,\) 
